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Recent trends in the graphene-based sensors for the detection of hydrogen peroxide

  • Received: 01 March 2018 Accepted: 03 May 2018 Published: 15 May 2018
  • This article intends to cover the latest progress and innovations in the field of graphene-based sensors for the detection of hydrogen peroxide (H2O2). The studies on the electrochemical behavior of a bioactive molecule have become one of the most rapidly developing fields. Biomedical engineering and biotechnology have an enthroned interest in fabricating more precise and accurate voltammetric/amperometric biosensors. One hastily growing area of biosensor design calls for the incorporation of carbon-based nano-materials such as two-dimensional graphene and its derivatives. Herein, a brief overview depicting the voltammetric techniques and how these techniques are useful in biosensing and sensing along with the details of surrounding important concepts such as sensitivity and limits of detection have been discussed in detail. The article discusses the graphene-based research for the effective immobilization of the enzymes such as horseradish peroxidase, hemoglobin, etc. for the accurate detection of H2O2 along with the detailed discussion on various material developed for the fabrication of non-enzymatic H2O2 sensors. The discussion ends with an outlook of future concepts that can be employed in sensor fabrication, as well as restrictions of already proposed materials and how such sensing can be improved. As such, this article can act as a roadmap to direct researchers in the direction of the next generation sensors highlighting the current advancements in the field.

    Citation: J Sharath Kumar, Naresh Chandra Murmu, Tapas Kuila. Recent trends in the graphene-based sensors for the detection of hydrogen peroxide[J]. AIMS Materials Science, 2018, 5(3): 422-466. doi: 10.3934/matersci.2018.3.422

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  • This article intends to cover the latest progress and innovations in the field of graphene-based sensors for the detection of hydrogen peroxide (H2O2). The studies on the electrochemical behavior of a bioactive molecule have become one of the most rapidly developing fields. Biomedical engineering and biotechnology have an enthroned interest in fabricating more precise and accurate voltammetric/amperometric biosensors. One hastily growing area of biosensor design calls for the incorporation of carbon-based nano-materials such as two-dimensional graphene and its derivatives. Herein, a brief overview depicting the voltammetric techniques and how these techniques are useful in biosensing and sensing along with the details of surrounding important concepts such as sensitivity and limits of detection have been discussed in detail. The article discusses the graphene-based research for the effective immobilization of the enzymes such as horseradish peroxidase, hemoglobin, etc. for the accurate detection of H2O2 along with the detailed discussion on various material developed for the fabrication of non-enzymatic H2O2 sensors. The discussion ends with an outlook of future concepts that can be employed in sensor fabrication, as well as restrictions of already proposed materials and how such sensing can be improved. As such, this article can act as a roadmap to direct researchers in the direction of the next generation sensors highlighting the current advancements in the field.


    We consider only simple and undirected labeled graphs and follow the terminology and notation of [5]. Let G=(V,E) be a graph and v be a vertex of G, we write NG(v) to denote the (open) neighborhood of v. The closed neighborhood of v is defined as NG[v]=NG(v){v}. The degree of v is defined as dG(v)=|NG(v)|. For UV, we write G[U] to denote the subgraph induced by U. The subgraph G[VU] is denoted by GU. Furthermore, GU can be written by Gu if U={u}.

    In a graph G, an independent set is a subset of vertices spanning no edges. Finding an independent set of maximum cardinality in a graph is a widely studied well-known problem of graph theory and is called the maximum independent set problem. In 1982, Hammer et al. [11] studied the maximum independent problem from different perspectives and investigated the vertices contained in all or in no maximum independent sets of a graph. Since then, researchers have extensively studied this kind of problem for some other vertex subsets with given properties. For example, Mynhardt [15], Cockayne et al. [10], and Blidia et al. [3] considered this kind of problem for minimum dominating sets, total dominating sets, and minimum double dominating sets of trees, respectively. Recently, Bouquet et al. [6] studied this kind of problem of minimum dominating sets on claw-free graphs, chordal graphs, and triangle-free graphs.

    In a graph G, a dissociation set is a subset of vertices F such that the induced subgraph G[F] has maximum degree at most 1. A maximum dissociation set of G is a dissociation set of maximum cardinality. The dissociation number ψ(G) of a graph G is the cardinality of a maximum dissociation set of G. The concept of dissociation set was introduced by Yannakakis [21] in 1981 and is a natural generalization of independent set. The maximum dissociation set problem, to find a maximum dissociation set in a graph is NP-hard for many subclasses of graphs such as K1,4-free bipartite graphs, C4-free bipartite graphs with vertex degree at most 3, planar graphs with vertex degree at most 4, triangle-free graphs, line graphs, etc. (see [2,4,16,17]). On the other hand, the problem is polynomially solvable for chordal and weakly chordal graphs, circular-arc graphs, AT-free graphs, (chair, K3)-free graphs, etc. (see [8,16]). In terms of exact algorithms, Kardoš, Katrenič and Schiermeyer [13] gave an O(1.5171n)-time exact algorithm for the problem in an n-vertex graph. Chang et al. [9] improved the result to O(1.4658n). Xiao and Kou [20] proposed an exact algorithm which can solve the problem in O(1.4656n) time and polynomial space or O(1.3659n) time and exponential space. Computing the dissociation number can be helpful in finding a lower bound for the 1-improper chromatic number of a graph [12]. And the maximum dissociation set problem has applications in telecommunications, scheduling, wireless sensor networks and networking security [1,7].

    A k-path vertex cover in a graph G is a subset of vertices intersecting every k-path of G, where a k-path is a path of order k. It is easy to see that a set S of vertices of a graph G is a 3-path vertex cover of G if and only if its complement V(G)S is a dissociation set of G. In this decade, the problem of finding a minimum k-path vertex cover in a graph has received great attention [7,13,14,19,20].

    The main purpose of this paper is to characterize the vertices contained in all maximum dissociation sets and in no maximum dissociation set of a tree. Define the vertex subsets A(G), F(G) and N(G) by

    A(G)={vV(G):v is in all maximum dissociation sets of G},F(G)={vV(G):v is in some but not all maximum dissociation sets of G},N(G)={vV(G):v is in no maximum dissociation set of G}.

    On the other hand, the study is also inspired by the relationship between the characteristic of vertex subsets A(G) and N(G) and the number of maximum dissociation sets in a graph G. In [18], Tu, Zhang and Shi found four structure theorems concerning the vertex subsets A(T) and N(T) for a tree T and determined the maximum number of maximum dissociation sets in a tree of order n.

    The paper is organized as follows. In Section 2, we introduce some necessary notation and lemmas. In Section 3, we characterize the vertex subsets A(T) and N(T) of a tree T. In Section 4, a linear recognition algorithm which can determine whether a given vertex in a tree is contained in all (or no) maximum dissociation sets of the tree will be presented. Thus, using the recognition algorithm, all vertices contained in all (or no) maximum dissociation sets of a tree of order n can be found in O(n2) time.

    A rooted tree T is a connected acyclic graph with a specified vertex r, called the root of T. Let T be a tree rooted at r and v be a vertex of T, each vertex on the path from the root r to the vertex v, including the vertex v itself, is called an ancestor of v, and a descendant of v is a vertex u such that v is an ancestor of u. An ancestor or descendant of a vertex is proper if it is not the vertex itself. The parent p(v) of v is the immediate proper ancestor of v, a child of v is a vertex u such that p(u)=v. Define the vertex subsets CT(v), DT(v), and DT[v] by

    CT(v)={uV(T):u is a child of v},
    DT(v)={uV(T):u is a proper descendant of v},
    DT[v]=D(v){v}.

    If no confusion occurs, these also be written by C(v), D(v), and D[v], respectively. We write Tv to denote the subtree induced by DT[v].

    A leaf in a tree is a vertex with degree 1, a branch vertex is a vertex with degree at least 3. We write B(T) to denote the set of branch vertices of T. A path P in T is called a vL path, if P joins v to a leaf that is a descendant of v. Denote the order of P by n(P), and for i=0,1,2, define

    Ci(v)={uC(v):Tu contains a uL path P with n(P)imod3}

    Now, some basic observations about maximum dissociation sets of the path are given.

    Observation 2.1. Let Pn=u1u2un be a path of order n3.

    (a) ψ(Pn)=2n+i3, where ni(mod3), i=0,1,2.

    (b) If n0(mod3), then there exists a maximum dissociation set of Pn that contains exactly one leaf.

    (c) If n1(mod3), then both leaves of Pn belong to all maximum dissociation sets of Pn, furthermore, Pn has a maximum dissociation set F such that dPn[F](u1)=0.

    (d) If n2(mod3), then there is only one maximum dissociation set F in Pn, furthermore, {u1,un}F and dPn[F](u1)=dPn[F](un)=1.

    Proof. (a) Since a maximum dissociation set of Pn contains at most two vertices of three consecutive vertices of Pn, one can easily check the truth of the statement by case analysis.

    (b) If n0(mod3), then F=V(Pn){u3i,1in3} is a maximum dissociation set of Pn that contains exactly one leaf.

    (c) If n1(mod3), then ψ(Pn)=2n+13. Suppose for a contradiction that there exists a maximum dissociation set in Pn that contains at most one leaf. Then ψ(Pn)=ψ(Pn1)=2(n1)3<2n+13, a contradiction. Thus both leaves of Pn belong to all maximum dissociation sets of Pn.

    Furthermore, F=V(Pn){u3i+2,0in43} is a maximum dissociation set of Pn such that dPn[F](u1)=0.

    (d) If n2(mod3), then ψ(Pn)=2n+23. Let F be any maximum dissociation set of Pn. If u1F, then ψ(Pn)=ψ(Pn1)=2(n1)+13<2n+23, a contradiction. If dPn[F](u1)=0, then ψ(Pn)=ψ(Pn2)+1=2(n2)3+1=2n13<2n+23, a contradiction. Thus, dPn[F](u1)=1. Similarly, dPn[F](un)=1. Since a maximum dissociation set of Pn contains at most two vertices of three consecutive vertices of Pn, one can easily check that there is only one maximum dissociation set in Pn.

    Firstly, we characterize A(T) and N(T) in the case where B(T)1.

    Lemma 2.2. Let T be a rooted tree with the root v. If for each uV(T){v}, dT(u)2, then

    ψ(T)={wC(v)ψ(Tw)+1,if|C2(v)|=0and|C1(v)|1;wC(v)ψ(Tw),otherwise.

    Proof. Because Tw is a path for each wC(v), it is easy to determine ψ(Tw) and Ci(v)Cj(v)= for ij. Note that wC(v)ψ(Tw)ψ(T)wC(v)ψ(Tw)+1. We consider the two cases.

    Case 1. |C2(v)|=0 and |C1(v)|1.

    If wC0(v), then TwPn with n0(mod3). Let Fw be a maximum dissociation set of Tw such that wFw (Fw exists by Observation 2.1(b)). If wC1(v), then TwPn with n1(mod3) and let Fw be a maximum dissociation set of Tw such that dTw[Fw](w)=0 (Fw exists by Observation 2.1(c)). Now, let

    F=wC(v)Fw{v}, (2.1)

    then F is a dissociation set of T and |F|=wC(v)ψ(Tw)+1. Thus, F is a maximum dissociation set of T and ψ(T)=wC(v)ψ(Tw)+1.

    Case 2. |C2(v)|1 or |C1(v)|2.

    Suppose for a contradiction that ψ(T)=wC(v)ψ(Tw)+1. Let F be a maximum dissociation set of T. Then, vF and FTw is a maximum dissociation set of Tw for each wC(v). Let Fw:=FTw for each wC(v). If wC2(v), then TwPn with n2(mod3). By Observation 2.1(d), wFw and dTw[Fw](w)=1. Thus, if |C2(v)|1, there is a 3-path in T[F] that contains the vertex v, a contradiction.

    If wC1(v), then TwPn with n1(mod3). By Observation 2.1(c), we have wFw. Thus, if |C1(v)|2, then there is a 3-path in T[F] that contains the vertex v, a contradiction.

    The proof is complete.

    Theorem 2.3. Let T be a rooted tree with the root v. If for each uV(T){v}, dT(u)2, then

    (a) vA(T) if and only if |C2(v)|=0 and |C1(v)|1;

    (b) vN(T) if and only if |C2(v)|=2 or |C1(v)|+|C2(v)|3.

    Proof. (a) Necessity. Suppose for a contradiction that |C2(v)|1 or |C1(v)|2 and vA(T). Let F=wC(v)Fw, where Fw is a maximum dissociation set of Tw for each wC(v). By Lemma 2.2, we have ψ(T)=wC(v)ψ(Tw). Thus, F is a maximum dissociation set of T and vF, which contradicts with vA(T).

    Sufficiency. Suppose that |C2(v)|=0 and |C1(v)|1. Then ψ(T)=wC(v)ψ(Tw)+1 by Lemma 2.2 and the vertex v is in all maximum dissociation sets of T. Thus, vA(T).

    The proof of (a) is complete.

    (b) Necessity. Suppose for a contradiction that |C2(v)|2 and |C1(v)|+|C2(v)|2 and vN(T).

    If |C2(v)|=0 and |C1(v)|1, then vA(T) by (a), a contradiction.

    If |C2(v)|=0 and |C1(v)|=2, then we assume w1,w2C1(v). For each wC0(v), there exists a maximum dissociation set Fw of Tw such that wFw by Observation 2.1(b). For w1C1(v), we have Tw1w1Pn with n0(mod3). Let Fw1 be a maximum dissociation set of Tw1w1. Then |Fw1|=ψ(Tw1)1 by Observation 2.1(a). For w2C1(v), there exists a maximum dissociation set Fw2 of Tw2 such that dTw2[Fw2](w2)=0 by Observation 2.1(c). Let F=wC0(v)FwFw1Fw2{v}, then F is a dissociation set of T and |F|=wC(v)ψ(Tw). By Lemma 2.2, F is a maximum dissociation set of T and vF, a contradiction.

    If |C2(v)|=1 and |C1(v)|1, then we assume w3C2(v). For w3C2(v), we have Tw3w3Pn with n1(mod3). Let Fw3 be a maximum dissociation set of Tw3w3, then |Fw3|=ψ(Tw3)1 by Observation 2.1(a). For each wC0(v), let Fw be a maximum dissociation set of Tw such that wFw. For wC1(v), let Fw be a maximum dissociation set of Tw such that dTw[Fw](w)=0. Now let F=wC(v){w3}FwFw3{v}, then F is a dissociation set of T and |F|=wC(v)ψ(Tw). By Lemma 2.2, F is a maximum dissociation set of T and vF, a contradiction.

    Sufficiency. Suppose for a contradiction that |C2(v)|=2 or |C1(v)|+|C2(v)|3, and vN(T). Let F be a maximum dissociation set of T such that vF. For each wC0(v), |FTw|ψ(Tw). If wC2(v), then TwPn with n2(mod3). Since vF, we have |FTw|ψ(Tw)1. If wC1(v), then every maximum dissociation set Fw of Tw contains the vertex w. Since vF, there are at least |C1(v)|1 vertices w in C1(v) such that |FTw|ψ(Tw)1. Hence, it is easy to check |F|<wC(v)ψ(Tw)ψ(T), a contradiction.

    The proof of (b) is complete.

    A technique called pruning process was introduced in [15]. Using the technique and Theorem 2.3, we can characterize A(T) and N(T) for an arbitrary tree T.

    Let T be a rooted tree with the root v and u be a branch vertex of T at maximum distance from v. It is easy to see that |C(u)|2 and dT(x)2 for each xD(u). If uv, we execute the following pruning process:

    ● if |C2(u)|1 or |C1(u)|2, then delete D[u],

    ● if |C2(u)|=0 and |C1(u)|1, then for all wC(u){z}, delete D[w], where z is the vertex in C1(u) if |C1(u)|=1, otherwise z is any one vertex in C(u).

    This step of pruning process is called a pruning of T at u. Repeat the above pruning process, finally, we obtain a unique tree ˉT called the pruning of T such that dˉT(u)2 for each uV(ˉT){v}. We will show that the root v is in all maximum dissociation sets (or in no maximum dissociation set) of T if and only if it is in all maximum dissociation sets (or in no maximum dissociation set) of the pruning ˉT of T.

    Lemma 3.1. Let T be a rooted tree with the root v and ˉT be the pruning of T. For every maximum dissociation set ˉF of ˉT, there exists a maximum dissociation set F of T such that vF if and only if vˉF. Conversely, for every maximum dissociation set F of T, there exists a maximum dissociation set ˉF of ˉT such that vˉF if and only if vF.

    Proof. We prove the lemma by induction on |B(T)|, where B(T)={uV(T){v}:d(u)3}. If |B(T)|=0, then T=ˉT and the result holds clearly. Suppose that when |B(T)|<k, the lemma holds. Let T be a tree with |B(T)|=k and u be a vertex of B(T) at maximum distance from v. Let T be the tree obtained from T by applying a pruning of T at u. Thus, ˉT is also the pruning of T.

    First, we show that for every maximum dissociation set ˉF of ˉT, there exists a maximum dissociation set F of T such that vF if and only if vˉF. By the induction hypothesis, for every maximum dissociation set ˉF of ˉT, there exists a maximum dissociation set F of T such that vF if and only if vˉF.

    We consider the following two cases.

    Case 1. |C2(u)|=0 and |C1(u)|1.

    For each wC(u){z}, TwPn with n0 (mod3). Let Fw be a maximum dissociation set of Tw such that wFw (Fw exists by Observation 2.1(b)). Let F=wC(u){z}FwF. Clearly, F is a maximum dissociation set of T. Since vF if and only if vF, we have vF if and only if vˉF.

    Case 2. |C2(u)|1 or |C1(u)|2.

    In this case, T=TD[u]. For each wC(u), let Fw be a maximum dissociation set of Tw. Let F=wC(u)FwF. Since |C2(u)|1 or |C1(u)|2, by Lemma 2.2, ψ(Tu)=wC(u)ψ(Tw)=wC(u)|Fw|, which implies that wC(u)Fw is a maximum dissociation set of Tu. Thus, F is a maximum dissociation set of T. Since vF if and only if vF, we have vF if and only if vˉF.

    Now, we show that for every maximum dissociation set F of T, there exists a maximum dissociation set ˉF of ˉT such that vˉF if and only if vF. Likewise, we consider the following two cases.

    Case 1. |C2(u)|=0 and |C1(u)|1.

    For each wC(u){z}, we have TwPn with n0(mod3). Let Fw be a maximum dissociation set of Tw such that wFw (Fw exists by Observation 2.1(b)).

    For every maximum dissociation set F of T, let F=FwC(u){z}D[w]. Then F is a dissociation set of T and vF if and only if vF. We will prove that F is a maximum dissociation set of T. Suppose for a contradiction that F1 is a maximum dissociation set of T with |F1|>|F|. Let F2=wC(u){z}FwF1. Then F2 is a dissociation set of T and

    |F2|=|F1|+wC(u){z}|Fw|>|F|+wC(u){z}|FD[w]|=|F|, (3.1)

    which contradicts the fact that F is a maximum dissociation set of T. Thus F is a maximum dissociation set of T. By the induction hypothesis, there exists a maximum dissociation set ˉF of ˉT such that vF if and only if vˉF. Hence, there exists a maximum dissociation set ˉF of ˉT such that vF if and only if vˉF.

    Case 2. |C2(u)|1 or |C1(u)|2.

    For each wC(u), let Fw be a maximum dissociation set of Tw. By Lemma 2.2, ψ(Tu)=wC(u)ψ(Tw)=wC(u)|Fw|.

    For every maximum dissociation set F of T, let F=FD[u]. We will prove that F is a maximum dissociation set of T. Suppose for a contradiction that F1 is a maximum dissociation set of T with |F1|>|F|. Let F2=wC(u)FwF1. Then F2 is a dissociation set of T and

    |F2|=|F1|+wC(u)|Fw|>|F|+|FD[u]|=|F|, (3.2)

    which contradicts the fact that F is a maximum dissociation set of T. Thus F is a maximum dissociation set of T. By the induction hypothesis, there exists a maximum dissociation set ˉF of ˉT such that vF if and only if vˉF. Thus, there exists a maximum dissociation set ˉF of ˉT such that vF if and only if vˉF.

    We complete the proof.

    By Lemma 3.1, we can obtain the following corollary.

    Corollary 3.2. Let T be a rooted tree with the root v and ˉT be the pruning of T, then vA(T) (or N(T)) if and only if vA(ˉT) (or N(ˉT)).

    By Theorem 2.3 and Corollary 3.2, the characterizations of A(T) and N(T) can be obtained immediately.

    Theorem 3.3. Let T be a tree and v be a vertex of T. Let Tv be the rooted tree obtained from T with the root v and ¯Tv be the pruning of Tv. Then

    (a) vA(T) if and only if |C2¯Tv(v)|=0 and |C1¯Tv(v)|1;

    (b) vN(T) if and only if |C2¯Tv(v)|=2 or |C1¯Tv(v)|+|C2¯Tv(v)|3.

    In [7], the authors presented an algorithm which can find a minimum k-path vertex cover in a tree in linear time, it follows that the problem of finding a maximum dissociation set of a tree can be solved in linear time. Let T be a tree and v a vertex of T. The equality ψ(Tv)=ψ(T)1 holds if and only if v belongs to all maximum dissociation sets of T. Thus the algorithm presented in [7] can be used to determine whether a given vertex in a tree belongs to all maximum dissociation sets of the tree. But, if ψ(Tv)=ψ(T), then it may be that v belongs to no maximum dissociation set of T, it may be that v belongs to some maximum dissociation sets but not all. Hence, the algorithm presented in [7] cannot be used to determine whether a given vertex in a tree belongs to no maximum dissociation set of the tree.

    In this section, we present a linear time recognition algorithm which can determine whether a given vertex in a tree is in all maximum dissociation sets, or in some maximum dissociation sets but not all, or in no maximum dissociation set. See Table 1.

    Table 1.  A recognition algorithm.
    Input: a tree T and a vertex vT.
    Output: vA(T); or vF(T); or vN(T).
       1. change the tree T into a rooted tree by choosing the vertex v as the root
       2. compute the distance d(v,u) from v to each other vertex u
       3. let B=B(T){v}, where B(T) is the set of branch vertices of T
       4. while B do
          4.1. choose a vertex u in B such that d(v,u) is maximum
          4.2. if |C2(u)|1 or |C1(u)|2, then
             TTD[u] and BB{u}
          4.3. else if |C1(u)|=1, then
             TTwC(u){z}D[w] and BB{u}, where z is the vertex in C1(u)
          4.4. else
             choose any one vertex z in C(u) and
            TTwC(u){z}D[w] and BB{u}
       5. If |C2(v)|=0 and |C1(v)|1, then
          output vA(T)
       6. else if |C2(v)|=2 or |C1(v)|+|C2(v)|3, then
          output vN(T)
       7. else
          output vF(T)

     | Show Table
    DownLoad: CSV

    For a rooted tree, we can use the breadth-first search algorithm to find the distance from the root to each other vertex in linear time. Step 4 is the pruning process of the rooted tree T. The cardinality of the set B is at most |V(T)| and will decrease by 1 after each loop execution. More precisely, let u be a vertex in B such that d(u,v) is maximum and w be a child of u, there exists only one wL path that joins w to a leaf that is a descendant of v and the order of the wL path can be obtained from the distances from the root to the vertex w and the leaf. Thus, we can compute the sets C2(u) and C1(u) in O(d(u)) time. The total runtime of Step 4 is O(uBd(u)), which implies that Step 4 can be executed in linear time. Thus, the runtime of the recognition algorithm is linear. Using the recognition algorithm we can find all vertices contained in all (or no) maximum dissociation sets of a tree of order n in O(n2) time.

    In this paper, we study the maximum dissociation set problem from different perspectives and characterize the vertices belonging to all maximum dissociation sets, and no maximum dissociation set of a tree. Based on the characterization, we present a linear time recognition algorithm which can determine whether a given vertex in a tree is in all maximum dissociation sets, or in some maximum dissociation sets but not all, or in no maximum dissociation set. Thus for a tree with n vertices, we can find all vertices belonging to all (or no) maximum dissociation sets of the tree in O(n2) time.

    The work is supported by Research Foundation for Advanced Talents of Beijing Technology and Business University (No. 19008021187). We would like to thank two anonymous referees for their many helpful suggestions and comments.

    The authors declare that they have no competing interests.

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